STAT 350: Lecture 8

I tried, as I was calculating things for the vectors , and to emphasize which things needed which assumptions.

So for instance we have following matrix identities which depend only on the model equation

where is the `hat' matrix ,

If we add the assumption that for each i then we get

and

If we add the assumption that the errors are homoscedastic ( for all i) and uncorrelated ( for all ) then we can compute variances and get

and

NOTE: usually we assume that the are independent and identically distributed which guarantees the homoscedastic, uncorrelated assumption above.

Next we add the assumption that the errors are independent normal variables. Then we conclude that each of , and have Multivariate Normal distributions with the means and variances as just described.